Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

We extend the definition of the classical Jacobi polynomials withindexes α,β > −1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the...

Optimal aggregate testing using Vandermonde polynomials and spectral methods.

Recycled aggregate (RA) has been used in various construction applications around the world mainly as sub-grade, roadwork and unbound materials, but not in higher-grade applications. The major barrier encountered is the variation of quality within RA, which causes lower strength, and poorer quality. This work studies the relationships among six parameters describing the characteristics of RA: (...

A Family of Generalized Jacobi Polynomials

The family of orthogonal polynomials corresponding to a generalized Jacobi weight function was considered by Wheeler and Gautschi who derived recurrence relations, both for the related Chebyshev moments and for the associated orthogonal polynomials. We obtain an explicit representation of these polynomials, from which the recurrence relation can be derived.

Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes α,β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the gener...

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials